Bertrand's theorem

Joseph Bertrand's theorem states that, among smooth attractive central potentials admitting stable circular orbits, requiring every bound orbit to be closed singles out the inverse-distance potential energy and the isotropic quadratic potential energy, up to additive constants. Rational radial-to-angular frequency ratios near a circular orbit provide only a necessary local closure test; finite-amplitude closure is the additional content. In $U=B{r}^n$, the radial-to-angular frequency ratio near a stable circle is $\sqrt{n+2}$, and stability requires $nB>0$ and $n>-2$. The all-bound-orbit cases are $n=-1$ and $n=2$.